When a cubic equation has a zero discriminant, and thus has real roots with multiplicity, is there any closed-form solution available for such roots? All sources I found just mention either the Cardano-method roots (which involve complex numbers even if the result is real), or the Viète/Descartes trigonometric form for the real roots, but they require a negative non-zero discriminant (if the discriminant is zero, you get an indetermination at the arccos of infinity).
Is there any closed-form (without complex numbers) for the roots when the discriminant is zero?
For the cubic equation $$ax^3+bx^2+cx+d=0$$ if $\Delta=0$ and $\Delta_0=b^2-3ac=0$ are equal to $0$, the equation has a triple root $$x=-\frac{b}{3a}$$ If $\Delta=0$ and $\Delta_0\neq 0$ , there are a double root
$$x_{1,2}=\frac{9ad-bc}{2(b^2-3ac)}$$ and a simple root, $$x_3=\frac{4abc-9a^2d-b^3}{a(b^2-3ac)}$$
Have look here.